Mathematical aspects of fullerenes

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Mathematical aspects of fullerenesVesna Andova, František Kardoš, Riste Škrekovski

To cite this version:Vesna Andova, František Kardoš, Riste Škrekovski. Mathematical aspects of fullerenes. Ars Mathe-matica Contemporanea, DMFA Slovenije, 2016, 11, pp.353 - 379. hal-01416354

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Also available at http://amc-journal.euISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)

ARS MATHEMATICA CONTEMPORANEA 11 (2016) 353–379

Mathematical aspects of fullerenes

Vesna Andova ∗

Faculty of Electrical Engineering and Information Technologies,Ss Cyril and Methodius Univ., Ruger Boskovik 18, 1000 Skopje, Macedonia

Frantisek KardosLaBRI, University of Bordeaux,

351, cours de la Liberation, 33405 Talence, France

Riste Skrekovski ∗

Faculty of Mathematics and Physics, University of Ljubljana,Jadranska 21, 1000 Ljubljana, Slovenia,

Faculty of Information Studies, 8000 Novo Mesto, Slovenia,Faculty of Mathematics, Natural Sciences and Information Technologies,

University of Primorska, Glagoljaska 8, 6000 Koper, Slovenia

Received 9 April 2015, accepted 8 February 2016, published online 25 March 2016

Abstract

Fullerene graphs are cubic, 3-connected, planar graphs with exactly 12 pentagonalfaces, while all other faces are hexagons. Fullerene graphs are mathematical models offullerene molecules, i.e., molecules comprised only by carbon atoms different than graphi-tes and diamonds. We give a survey on fullerene graphs from our perspective, which couldbe also considered as an introduction to this topic. Different types of fullerene graphs areconsidered, their symmetries, and construction methods. We give an overview of somegraph invariants that can possibly correlate with the fullerene molecule stability, such as:the bipartite edge frustration, the independence number, the saturation number, the numberof perfect matchings, etc.

Keywords: Fullerene, cubic graph, planar graph, topological indices.

Math. Subj. Class.: 05C12, 05C90, 92E10, 94C12

∗Partially supported by Slovenian ARRS Program P1-00383 and Creative Core - FISNM - 3330-13-500033.E-mail addresses: [email protected] (Vesna Andova), [email protected] (Frantisek Kardos),

[email protected] (Riste Skrekovski)

cb This work is licensed under http://creativecommons.org/licenses/by/3.0/

354 Ars Math. Contemp. 11 (2016) 353–379

1 IntroductionThe first fullerene molecule, with a structure like a football, was discovered experimen-tally in 1985 by Kroto et al. [75]. The discovered molecule C60, is comprised only by 60carbon atoms, and it resembles the Richard Buckminster Fuller’s geodetic dome, thereforeit was named buckminsterfullerene. Until that moment the only all-carbon structures themodern science was aware of were graphite and diamond. In 1991, the Science magazinepronounced the buckminsterfullerene for the “Molecule of the year”, and later in 1996 thediscovery of C60 was rewarded with the Nobel price for chemistry. Soon after the exper-imental discovery of the buckminsterfullerene, its existence in the nature was confirmedalong with similar structures having 70, 76, 78, 82, 84, 90, 94, or 96 carbon atoms. Eachof these all-carbon molecules have polyhedral structure, and all faces of the polyhedron areeither pentagons or hexagons. All polyhedral molecules made entirely of carbon atoms arecalled fullerenes. The discovery of the Buckminsterfullerene marked the birth of fullerenechemistry and nanotechnology, but at the same time the fullerenes were studied from dif-ferent perspectives. The experimental work was paralleled by theoretical investigations,applying the methods of graph theory to the mathematical models of fullerene moleculescalled fullerene graphs.

The study from graph theoretical point of view has been motivated by a search forinvariants that will correlate with their stability as a chemical compound. Later graph in-variant were used in order to predict the physical and chemical properties of a fullerenecompound. A number of graph-theoretical invariants were examined as potential stabilitypredictors with various degrees of success [42, 29]. Graph theory invariants that have beenconsidered as possible stability predictors are: the bipartite edge frustration, the indepen-dence number, the saturation number, the number of perfect matchings, etc. As a result ofthose investigations, we have achieved a fairly thorough understanding of fullerene graphsand their properties. However, some problems and questions still remain open [25, 78, 49].Special place among them have several interesting conjectures made by Graffiti, a conjec-ture making software [41].

2 Fullerene graphsFullerenes can also be seen as graphs, vertices represent atoms, and edges represent bondsbetween atoms. A fullerene graph is a 3-connected 3-regular planar graph with only pen-tagonal and hexagonal faces. In what follows fullerene graphs will be also called fullerenes.Due to Whitney’s Theorem (1933) simple planar 3-connected graphs have a unique planarembedding, and therefore the same holds for fullerene graphs. By Euler’s formula followsthe next property of fullerene graphs.

Proposition 2.1. The number of pentagonal faces in a fullerene graph is 12.

The previous result gives no restriction on the number of hexagons. Grunbaum andMotzkin [59] showed that fullerene graphs exist for any number of hexagonal faces exceptfor 1, i.e., they proved the following.

Theorem 2.2. Fullerene graphs with n vertices exist for all even n ≥ 24 and for n = 20.

Usually in chemistry the fullerene molecule on n vertices is denoted by Cn. Althoughthe number of pentagonal faces is negligible compared to the number of hexagonal faces,their layout is crucial for the shape of the corresponding fullerene molecule. Notice that as

V. Andova, F. Kardos, R. Skrekovski: Mathematical aspects of fullerenes 355

the number of vertices (hexagons) grows, the number of fullerenes increases as well. Forexample the fullerenes on 20, 24 and 26 vertices have the unique layout, but the fullereneC40 has 40 isomers, while the buckminsterfullerene has 1812 different isomers. Thereis a believe that number of fullerenes on n vertices is of order Θ(n9), see Fowler andManolopoulos in [45] and Cioslowski [22] for more details.

2.1 Types of fullerene graphs

Regarding the position of the pentagons we distinguish several types of fullerene graphs.The fullerene graphs where no two pentagons are adjacent, i.e., each pentagon is sur-rounded by five hexagons, satisfy the isolated pentagon rule or shortly IPR, and they areconsidered as stable fullerene compounds [76]. Cioslowski [22] stated the following con-jecture concerning the number of IPR fullerenes on n vertices.

Conjecture 2.3. For all n > 106, the number of the IPR fullerene isomers with n carbonatoms is bracketed by the total numbers of isomers of the Cn−50 and Cn−48 fullerenes.

If all pentagonal faces are “equally distributed”, we obtain fullerene graphs of icosahe-dral symmetry, whose smallest representative is the dodecahedron. The dodecahedron isthe only icosahedral fullerene that does not satisfy the IPR. On the other hand, if the pen-tagonal faces are grouped in two clusters by six, we obtain nanotubical fullerene graphs.Now, we consider these types of graphs separately.

2.1.1 Icosahedral fullerene graphs

The common feature of all icosahedral fullerenes is their geometrical shape. The simplesticosahedral fullerene graph is the dodecahedron, C20; the next one is the famous buckmin-sterfullerene C60. Caspar and Klug [21] and Coxeter [23] suggested a method that con-structs the duals of the icosahedral fullerene graphs: geodesic domes, i.e., triangulations ofthe sphere with vertices of degree 5 and 6.

A

C

B

~G = (2, 3)

Figure 1: Construction of a (2, 3)-triangle, a metaface of a (2, 3)-icosahedral fullerenegraph. The vertices of the equilateral triangle ABC are centers of pentagons.

Goldberg [54] observed that all icosahedral fullerene graphs can be obtained by map-ping (a part of) the hexagonal grid onto the triangular faces of an icosahedron. He alsoshowed that the number of vertices n in a fullerene of icosahedral symmetry can be deter-mined by two integers i and j by the following equation, conveniently called the Goldbergequation

n = 20(i2 + ij + j2). (2.1)

356 Ars Math. Contemp. 11 (2016) 353–379

The integers i and j in the Goldberg equation are in fact the components of a two-dimensional Goldberg vector ~G = (i, j), sometimes also called Coxeter coordinates. Wealways assume that 0 ≤ i ≤ j and 0 < j in order to avoid the mirror effect. This vectordetermines the distance and positions of the vertices of the (i, j)-triangle in the hexagonallattice, see Figure 1 for a construction method of an (2, 3)-triangle. Precisely 20 such(i, j)-triangles produce an (i, j)-icosahedral fullerene in a manner shown on Figure 2. Thevertices of the triangles are centers of the 12 pentagons of the fullerene graph. Observe thatthe dodecahedron is the (0, 1)-icosahedral fullerene, whereas the buckminsterfullerene isthe (1, 1)-icosahedral fullerene.

If i = j or j = 0, then the (i, j)-icosahedral fullerene graph has mirror symmetry, itssymmetry group is (isomorphic to) Ih, the full symmetry group of a regular icosahedron.Otherwise, its symmetry group is (isomorphic to) I , the group of rotations of a regularicosahedron.

A A A A A

CB B

D D

E E E E E

B′ C ′

Figure 2: A (2, 3)-icosahedral fullerene. Its triangular faces are constructed as on Figure 1.The vertices with the same label coincide.

2.1.2 Fullerenes with other symmetry groups

A symmetry group of a fullerene can contain an m-fold rotational axis only for m = 2,3, 5, and 6. This makes the list of all possible symmetry groups finite; it consists of 36groups:

icosahedral: Ih,I ;

tetrahedral: Th,Td,T ;

prismatic: D6h,D5h,D3h,D2h;

antiprismatic: D6d,D5d,D3d,D2d;


dihedral: D6,D5,D3,D2;

skewed: S12,S10,S6,S4;

rotation-reflection: C6h,C5h,C3h,C2h;

pyramidal: C6v,C5v,C3v,C2v;

rotation only: C6,C5,C3,C2,

nonaxial: Cs,Ci,C1.

D6h D5h D3h D2h

D6d D5d D3d D2d

D6 D5 D3 D2

Figure 3: Examples of smallest fullerenes with prismatic, antiprismatic and dihedral sym-metry groups.

It was proved that whenever a five-fold or six-fold rotational axis is present, the struc-ture of the fullerene implies a perpendicular twofold rotational axis [46]. This means thatthe groups S12,S10,C6h,C5h,C6v,C5v,C6, and C5 only occur as subgroups of icosahe-dral, prismatic, antiprismatic, or dihedral groups Ih,I ,D6h,D5h,D6d,D5d,D6, and D5.The final list of 28 fullerene symmetry groups is therefore:

Ih,I ; Th,Td,T ; D6h,D5h,D3h,D2h; D6d,D5d,D3d,D2d;

D6,D5,D3,D2; S6,S4; C3h,C2h; C3v,C2v; C3,C2; Cs,Ci,C1.

As shown in [46], fullerenes of each group can be found among all fullerenes on nvertices 20 ≤ n ≤ 140. From a fullerene with a given symmetry group, by a leapfrog

358 Ars Math. Contemp. 11 (2016) 353–379

Th Td T

Figure 4: Examples of smallest fullerenes with tetrahedral symmetry groups.

transformation (see Figure 9) one can construct a bigger fullerene with the same symmetrygroup. Thus, there is an isolated pentagon fullerene for each of the 28 symmetry groups.Symmetry groups for small fullerenes are considered in [8] and [45]. In the later work thesmallest fullerene is found for each possible symmetry group, see Figures 3, 4, and 5. Allfullerenes with symmetry group of at least 10 elements, i.e., Ih, I , Td, Th, T , D6h, D6d,D6, D5h, D5d or D5, are catalogued in [57].

2.1.3 Nanotubical fullerene graphs

While the icosahedral fullerenes have “spherical” shape, there is a class of fullerene graphsof tubular shapes, called nanotubical graphs or simply nanotubes. From the aspect ofmathematics they are not well defined. However, they are cylindrical in shape, with thetwo ends capped with a subgraph containing six pentagons and possibly some hexagons.The cylindrical part of a nanotube can be obtained by rolling a planar hexagonal grid. Theway the grid is wrapped is represented by a Goldberg vector (i, j), called also the type ofthe nanotube. See Figure 6 for an example of the construction of the cylindrical part of ananotube, also called an open nanotube.

For a given nanotube of type (i, j), the sum i + j is called perimeter of the nanotube.The smallest possible perimeter is 1, but the open nanotube (0, 1) is not simple. The opennanotubes with perimeter 2 are the nanotubes (1, 1) and (0, 2), but these graphs are not 3-connected. The smallest perimeter such that the graph is 3-connected is 3. On the other sidein nature, the thinnest (open) nanotube is (2, 2)-nanotube [95]. Since fullerene graphs arecyclically 5-edge connected [28] it is not possible to close an open nanotube with perimetersmaller than 5. Even more there is a cap with six pentagons that closes (0, 5)-nanotube, butthere are no caps for (1, 4), and (2, 3)-nanotube. For all vectors (i, j) with i+ j ≥ 6 thereexists a nanotube. A (0, 5)-nanotube is depicted on Figure 13.

Notice that the caps of nanotubes are not well defined. One can find (infinitely) manycaps for an (i, j)-nanotube by inserting or removing a hexagonal face from a patch withsix pentagons that is a subgraph of the nanotube. On the other hand Brinkmann et al. [15]showed that from all possible caps for an (i, j)-nanotube, we can always choose a “nice”cap. We call a cap P nice if its boundary can be represented by the sequence (23)i(32)j ,and at least one pentagon is incident to the boundary of P . An edge on the boundaryis called i-j edge if its end vertices are of degree i and j, respectively. Observe that if


S6 S4 C3h C2h

C3v C2v C3 C2

Cs Ci C1

Figure 5: Examples of smallest fullerenes with other symmetry groups.

i ≥ j > 0 the boundary of a nice cap has precisely one 2-2 edge and one 3-3 edge. Ifj = 0, all the edges on the boundary are 2-3 edges. Starting with any cap, and inserting orremoving a finitely many hexagons we can construct a nice cap. If the number of verticesof the nanotube is large enough compared to i and j, we can always choose the two capssuch that they are nice and disjoint.

Although the caps are not well defined, a result from Brinkmann et al. [15] provides theexistence of “nice” caps.

Theorem 2.4. Let F be a long enough (i, j)-nanotube. Then, among all possible caps ofF , there are caps such that their border can be represented by the sequence (23)i(32)j .

Let P(i, j) be the set of all possible nice caps for an (i, j)-nanotube. A nice cap ac-cording to Brinkmann et al. [15] can be transformed into a nice cap according to currentdefinition by inserting less than i+ j hexagons. Brinkmann et al. in [15, 13] give a methodfor constructing all possible nice caps for an (i, j)-nanotubes. Due to this result we havethat P(i, j) is a finite set.

Nanotubes with i = 0 are called zig-zag nanotubes, and the ones with i = j are calledarmchair nanotubes. These are the only types of nanotubes where the cylindrical part has amirror symmetry. The buckminsterfullerene C60 can be viewed as the smallest nanotube oftype (5, 5), see Figure 7. It is also the smallest nanotube with the caps satisfying the IPR.

360 Ars Math. Contemp. 11 (2016) 353–379

A

A

p~a1

q~a2

p~a1 + q~a2

Figure 6: An example of a (2, 4) nanotube. The hexagons denoted equally overlap.

There are IPR nanotubes for all types (i, j) with i + j ≥ 11 and for (5, 5), (0, 9), (0, 10),(1, 9), and (2, 8).

=

+

+

Figure 7: Buckminsterfullerene is the smallest nanotube of type (5, 5).

2.2 Pentagonal-hexagonal patches

Not every fullerene graph is a nanotube. Therefore, subgraphs with similar structure as thenanotube caps were studied in order to facilitate fullerene generation and enumeration.

A patch is a 2-connected plane graph with only pentagonal or hexagonal faces, exceptmaybe one (the unbounded) face. All interior vertices of a patch are of degree 3, and allvertices of the exceptional face, which we consider as the outer face, are of degree 2 or 3.Let P be a patch with h hexagons and p pentagons. All the vertices incident to the outerface – the boundary of P , b(P ) – are of degree two or three; let the number of vertices ofdegree two and three on the border of P be denoted by n2,b(P ) and n3,b(P ), respectively.Clearly for the size of the border holds |b(P )| = n2,b(P ) + n3,b(P ). Bornhoft et al. [11],and independently Kardos et al. [70] give the following relation between the number ofvertices on the border and the number of pentagons in the patch.

Proposition 2.5. Let P be a patch with p pentagons. Then, |b(P )| = 2n3,b(P ) + 6− p.


The last result gives the relation between n2,b(P ) and n3,b(P ), i.e., n2,b(P )−n3,b(P )=6− p. If P is a patch with six pentagons, then the number of vertices of degree 2, and thenumber of vertices of degree 3 are equal, not depending on the size or shape of P . Observethat if one ring of hexagons around P is added, the length of the border does not change. IfP has less than six pentagons, then after adding a ring of hexagons around P , the numberof vertices on the outer face will increase, and vice versa if P has more than six pentagonsafter adding the ring of hexagons the number of vertices on the outer face will decrease.This result confirms that in order to preserve the constant perimeter of the tube in thenanotubes, each cap must have precisely six pentagons.

The degrees of the vertices on the boundary can be regarded as a sequence of length|b(P )| whose elements are 2 and 3.

Inserting a face (pentagon or hexagon) to a patch P means creating a new patch fromP with one face more than P such that the new face is incident to at least one edge fromthe outer face of P . Removing a face is an inverse operation of the insertion.

Let the set of all patches with h hexagons and p pentagons be denoted by Pp,h. There isa “monotone” method (with respect to the length of the patch boundary) for construction ofall possible patches with at most 5 pentagons and given upper bound on the border startingwith pentagonal or hexagonal face.

Proposition 2.6. Each patch P with h hexagons and p ≤ 5 pentagons can be obtained byadding a pentagon or hexagon to some patch P ′ ∈ Pp−1,h ∪ Pp,h−1 such that |b(P )| ≥|b(P ′)|.

Bornhoft et al. [11] determined the minimum and maximum possible boundary lengthsof pentagon-hexagon patches with h hexagons and p ≤ 6 pentagons. They show thatthe minimal boundary length is obtained for a face spiral patch Sp,h starting with all ppentagons and continuing with the hexagons. A maximum boundary length is obtainedfor a patch with a tree as its inner dual. Let min(p, h) and max(p, h) be the minimalboundary length and maximal boundary length of a patch with p pentagons and h hexagonsrespectively. The following theorem from [11] determines which intermediate values canoccur as a boundary length of a patch with p pentagons and h hexagons.

Theorem 2.7. For p ≤ 6 there exists a patch P with p pentagons and h hexagons and aboundary length b if and only if min(p, h) ≤ b ≤ max(p, h) and b ≡ p (mod 2).

3 Construction of fullerene graphsAs said before for a fullerene on n vertices, there should be Θ(n9) isomers. Searchingfor an efficient method that constructs all possible isomers on n vertices resulted in severaldifferent approaches. Here, we consider the spiral method, Stone-Wales rearrangement,and patch replacement transformation.

By Coxeter and Coldberg constructions, one can construct icosahedral fullerene as de-scribed in Section 2. A modification of the same method can be used in order to constructother fullerenes with smaller number of symmetries, but as the number of symmetries de-creases, this construction method gets more complicated. When icosahedral fullerenes areconstructed all 20 triangles are the same and equilateral, but in the case of fullerenes withlower symmetries the triangles can be isosceles or scalene. This implies that the trianglesare not equal, and therefore more parameters are needed, what makes this method compli-

362 Ars Math. Contemp. 11 (2016) 353–379

cated. A Coxeter method for construction of tetrahedral and dihedral fullerenes is givenin [80].

Fullerenes can be constructed from other fullerenes with two simple techniques; rear-ranging the faces known as Stone-Wales rearrangement, or by adding two new vertices byEndo-Kroto insertion:

• The Stone-Wales rearrangement [90] is a transformation that constructs an isomer ofa fullerene graph. This transformation rearranges the pentagons and hexagons in apatch with two pentagons and two hexagons without changing the number of verticesas shown on Figure 8.

• The Endo-Kroto C2 insertion [38] is a transformation that results a bigger fullerene,i.e., a fullerene on n+2 vertices. This transformation is possible only if the fullerenecontains a patch as on Figure 8 (c).

(a) (b)

(c) (d)

Stone-Wales rearrangement Endo-Kroto C2 insertion

Figure 8: Stone-Wales rearrangement and Endo-Kroto C2 insertion. Stone-Wales rear-rangement: replaces the patch (a) by the patch (b). Endo-Kroto C2 insertion: replaces thepatch (c) by the patch (d), and inserts two new vertices.

It is easy to see that these two transformations do not allow to construct all the fullerenegraphs. Moreover, since the pentagons can be arbitrarily far away from each other, nofinite set of transformations can be sufficient to generate all the fullerene graphs. However,later it was shown [47] that these two transformations are special cases of more generaltransformations based on patch replacement considered later in Subsection 3.2.

Besides these methods, the leapfrog operation of a fullerene graph is another methodthat produces (bigger) fullerenes. The leapfrog operation of a fullerene graph G, L(G),is usually used for construction of bigger and isolated pentagon fullerenes. The leapfroggraph L(G) is obtained by truncating the dual of the fullerene graph G, i.e.,

L(G) = Tr(Du(G)) ,

where Du(G) is dualization, and Tr(G) is truncation of the graph G. At the same timeleapfrog operation can be viewed as a composition of stelliation St(G), and dualizationDu(G),

L(G) = Du(St(G)) .

The operations stellisation, dualization and truncation for a given graph are defined asfollows:

• Stellisation, St(G), adds a vertex in the center of each face of a planar graph G, andconnects the new vertex with each boundary vertex of the corresponding face. Noticethat this operation is also a triangulation.


(a) (b)

Figure 9: (a) Dodecahedron and its dual graph. (b) Buckminsterfullerene graph obtainedwith the leapfrog operation on the dodecahedron, truncating dodecahedron’s dual.

• Dualization of a graph G is the operation that produces the dual of G. Recall thatthe dual graph of a planar graph G, Du(G), is a graph with vertices correspondingto each face of G. Two vertices in Du(G) are adjacent if the corresponding faces inG share an edge.

• Truncation, Tr(G), of a plane graph G is an operation that adds two new vertices oneach edge, and then removes the vertices of G. Two vertices are adjacent if they areadded to the same edge of G or they belong to two consecutive edges incident to thesame vertex of G. The truncation of a polyhedron cuts off one third of each edge ateach of both ends.

It is easy to see that L(G) is a fullerene graph on |2E(G)| vertices. Even more L(G) is anisolated pentagon fullerene. The leapfrog graph of the dodecahedron C20 is the buckmin-sterfullerene C60, see Figure 9.

3.1 Spiral method

A face spiral is an ordering of the faces of the fullerene graph such that each face in thespiral has a common edge with its predecessor and successor on the spiral (except the firstand last faces). The face spiral ordering is shown in Figure 10. A face spiral sequenceis a sequence of twelve different integers that correspond to the position of the twelvepentagons in the fullerene. The first face in the face spiral has the position 0, see Figure 10.The face spiral sequence can be also be given as a sequence of twelve fives and a number ofsixes, and the position of the fives (resp. sixes) corresponds to the position of the pentagons(resp. hexagons) in the spiral ordering.

The spiral conjecture states that the surface of a fullerene polyhedron can be unwrappedas a continuous spiral strip of adjacent faces, i.e., every fullerene has a face spiral. Althougheach fullerene with at most 176 vertices has at least one face spiral [12] and fullerenes withicosahedral symmetry have a spiral [48], the conjecture is not true in general, since thereare unspiral fullerenes with more than 380 vertices [79].

364 Ars Math. Contemp. 11 (2016) 353–379

An analog to the face spiral is the vertex spiral. A spiral is a path that starts with anedge and always takes the most left (resp. right) edge not incident with a vertex that isalready on the spiral. A vertex spiral is the numbering of the fullerene vertices in the orderthey appear on the spiral. Although the vertex spiral is useful for the nomenclature of thefullerene isomers, it is shown that such paths do not always exist. One such example is the(2, 1)-icosahedral fullerene C80 [45].

Figure 10: The fullerene on 28 vertices is spiral. The face spiral sequence is 0, 2, 4, 5, 6, 7,8, 9, 10, 11, 12, 14, or 5, 6, 5, 6, 5, 5, 5, 5, 5, 5, 5, 5, 5, 6, 5.

A spiral method is a consequence of the spiral conjecture. This method simply gen-erates all the sequences of n

2 − 10 sixes and twelve fives as a face spiral of pentagonsand hexagons and tries to close them into a fullerene on n vertices. This method was thefirst attempt to construct all fullerenes on n vertices given by Manolopoulos and May [81].This algorithm is incomplete, i.e., not every fullerene can be generated with it, since not allfullerenes are spiral [79].

3.2 Expansion operations

Expansion is an operation that replaces a patch from the fullerene graph with a biggerone, and the inverse operation is called a reduction. These operations are also known aspatch replacement operations. The simplest patch replacement operations are Stone-Walesrearrangement and Endo-Kroto C2 insertion considered above.

According to the results of Brinkmann et al. [14], no finite set of expansion operationsis sufficient to construct all fullerene graphs, starting with a smaller fullerene. Hashem-inezhad et al. [62] defined an infinite set of expansions grouped in three classes: Li fori ≥ 0, Bi,j for i, j ≥ 0 and F expansion. These three expansions are defined as follows:

• Li expansion. This expansion is defined for i ≥ 0, and uses a path of length 2i + 3that alternate left-right at each of its vertices and connects two pentagonal faces, alsoknown as it zig-zag path. On Figure 11 the expansion L1 is shown. This expansionstarts with a subgraph as shown on Figure 11(a), between the two pentagonal facesthere is a ziz-zag path of length 5, and replaces this subgraph with a subgraph shownon Figure 11(b). The Li expansion adds i+ 2 new (hexagonal) faces, see Figure 11,i of which are between the two pentagonal faces.

• Bi,j expansion. Similarly as in the case of Li expansion, theBi,j , i, j ≥ 0 expansionuses a path of length 2i+ 2j + 5 between two pentagonal faces. This path alternates


f1 f2 f3g1 g2 g3

f1f2

f3

g1

g2g3

(a) (b)

Figure 11: L1 expansion.

left-right for all the vertices on the path except for the vertices 2i+2 and 2i+3, thesetwo vertices keep the same orientation, see Figure 12(a). The Bi,j expansion addsi+ j + 3 new (hexagonal) faces, i+ j + 1 of which are between the two pentagons.Figure 12 shows the expansionB0,1, the subgraph on Figure 12(a) is replaced by thesubgraph shown on Figure 12(b).

• F expansion. This expansion adds five hexagonal faces around a cap of a (5, 0)-nanotube (see Figure 13) comprised only by six pentagons one of which is central.This operation is equivalent to adding an extra ring in a (5, 0)-nanotube.

Observe that the faces drawn completely or the ones labeled by fk or gk are all distinct.Also, the faces that are not drawn completely can be either pentagons or hexagons.

f1f2

f3

f4

g1g2

g3

g4

f1f2

f3f4

g1

g2g3

g4

(a) (b)

Figure 12: B1,0 expansion.

Using this infinite class of expansions Hasheminezhad et al. showed [62] the following.

Theorem 3.1. Every fullerene except the tetrahedral C28 can be constructed from the do-decahedron C20 using expansions Li, Bi,j , and F . Moreover, if the fullerene is not a(5, 0)-nanotube, it can be constructed from C20 using expansions Li and Bi,j .

Additionally, they showed that all fullerene except the tetrahedral C28 (see Figure 4)and a (5, 0)-nanotube (see Figure 13) with up to 300 vertices can be constructed from thedodecahedron and Li expansion.

4 Structural properties of fullerene graphsAs we already mentioned earlier, the distribution of the twelve pentagons determine theshape of the fullerene and its structural properties. In this section, we consider severalstructural properties of fullerenes, such as: cyclic-edge connectivity, diameter, bipartisa-tion, independence number, specter, etc.

366 Ars Math. Contemp. 11 (2016) 353–379

4.1 Cyclic connectivity

Fullerene graphs are 3-connected and 3-edge connected, but not 4-connected and 4-edgeconnected. Here we consider cyclic connectivity. Recall that an edge-cut C of G is cyclicif each component of G− C contains a cycle. A graph is cyclically k-edge-connected if atleast k edges must be removed to disconnect it into two components such that each containsa cycle. A cyclical edge-cut is trivial if it isolates a single cycle.

A graph G is cyclically k-connected if G can be expressed as a union of two edgedisjoint subgraphsG1 andG2, bothG1 andG2 containing cycles, and |V (G1)∩V (G2)| ≥k. The cyclic connectivity ofG is the maximum integer k (if exists) such thatG is cyclicallyk-connected. For fullerene graphs, Doslic [28] proved the following theorem.

Theorem 4.1. Every fullerene graph F is cyclically 5-edge-connected.

Clearly, the result is best possible, since each pentagonal face is separated from the restof the graph by 5 edges. We say that a cyclic 5-edge cut is trivial, if it separates only onepentagonal face, otherwise it is not trivial. Kardos and Skrekovski [71], and Marusic andKutnar [77] proved that fullerenes which have nontrivial 5-edge-cut are of unique type.

Theorem 4.2. Nanotubes of type (0, 5) are the only fullerene graphs with a nontrivialcyclic 5-edge-cut.

A (0, 5)-nanotube is depicted on Figure 13. Observe that every nontrivial cyclic 5-edge-cut of a (0, 5)-nanotube separates the two caps (containing six pentagonal faces each)from each other.

Figure 13: A (0, 5)-nanotube.

Similarly, for each nanotube of type (i, j) there are cyclic (i+ j)-edge cuts separatingthe two caps from each other. However, the nanotubes are not the only fullerene graphswith such edge-cuts. An example of a fullerene graph, in which the twelve pentagonalfaces are partitioned into four clusters containing 1, 2, 4, and 5 pentagons each, is depictedon Figure 14. It is easy to see that this graph is not a nanotube.

With the next theorem Qi and Zhang [96] determined the cyclic connectivity of fuller-ene graphs.

Theorem 4.3. Every fullerene graph F is 5-cyclically connected.


Figure 14: A half of a fullerene graph, which is not a nanotube, but it is possible to separatesix pentagonal face from the other six. In order to obtain the whole graph, the depictedgraph is to be glued with its mirror copy along the boundary of the outer face such that the3-regularity is preserved.

4.2 Diameter of fullerene graphs

For large enough fullerene graphs of spherical shape, the diameter diam(F ), the largestdistance between any two vertices, is proportional to the radius of the sphere, whereas thenumber of vertices n is proportional to its surface. Hence, one could expect the diameter ofsuch graphs to be of order Θ(

√n). On the other hand, for a nanotubical fullerene graph of

type (i, j), the diameter diam(F ) is proportional to the length of the cylindrical part of thegraph, whereas the number of vertices n is proportional to the product of the length of thetube and its circumference i+ j. In this case, one could expect the diameter of such graphsis of order Θ(n).

A well known result on the degree-diameter problem states that the number of verticesin a planar graph with maximum degree 3 grows at most exponentially with diameter [50].Let G be a planar graph with maximum degree 3. Then, G has at most

2diam(G)+1 − 1

vertices. This results gives a logarithmic lower bound on the diameter in terms of thenumber of vertices. The logarithmic character of the bound can be attributed to the presenceof faces of large size. It would be reasonable to expect that better lower bounds exist forpolyhedral graphs with bounded face size.

As mentioned above, fullerene graphs only have pentagonal and hexagonal faces, andthis fact can be used to show [2] that if F is a fullerene graph on n vertices, then diam(F ) ≥√

24n− 15/6− 1/2.For fullerene graphs with full icosahedral symmetry the diameter has been determined

exactly in [6].

Theorem 4.4. The diameter of a fullerene graph with n vertices is of order Ω(√n). If F is

a full icosahedral symmetry fullerene (i, j), 0 ≤ i ≤ j, 0 < j, then diam(F ) = 4j+6j−1.

Observe that an icosahedral fullerene (i, j) has full icosahedral symmetry if i = 0 ori = j. By (2.1) we have that if i = 0, then diam(F ) =

√9n/5 − 1, and if i = j, then

diam(F ) =√

5n/3−1. As the geometrical shape of these fullerenes is “almost spherical”we believed that they have the smallest diameter, so we conjectured that the diameter of anyfullerene graph on n vertices is at least b

√5n/3c − 1. Nicodemos and Stehlık [84] found

an infinite class of fullerene graphs of diameter at most√

4n/3, where n is the number ofvertices, and disproved this conjecture.

368 Ars Math. Contemp. 11 (2016) 353–379

As far as the upper bound is concerned, the nanotubes of small circumference are theextremal graphs, as says the following theorem [2].

Theorem 4.5. Let F be a fullerene graph with n vertices that is not a (5, 0)-nanotube.Then,

diam(F ) ≤ n

6+

5

2.

If F is a (5, 0)-nanotube on n = 30, 40 vertices, then diam(F ) = n/5, and if n ≥ 50, thendiam(F ) = n/5− 1.

4.3 Bipartisation of fullerenes

The bipartite edge frustration of a graph G, denoted by ϕ(G), is the smallest cardinality ofa set of edges of G that needs to be removed from G in order to obtain a bipartite spanningsubgraph. Bipartite edge frustration of a fullerene graph G can be efficiently computed byfinding a minimum-weight perfect matching in the pentagon-distance graph of G [35]. Inthe same reference it was shown that ϕ(G) ≥ 6 for any fullerene graph G and that thisbound is sharp.

If the fullerene graph satisfies the isolated pentagon rule, then ϕ(G) ≥ 12. Further-more, it was shown that the bipartite edge frustration of fullerene graphs with icosahedralsymmetry is proportional to the square root of the number of vertices (see [35, Proposition11 and Corollary 12]). The numerical computations suggested that it cannot behave worsethan that, and prompted the authors conjecture that the independence number of a fullerenegraph on n vertices is at most

√12n/5.

First, Dvorak, Lidicky, and Skrekovski [37] proved a theorem with a weaker multi-plicative constant. Later Faria, Klein and Stehlık [43] proved the following celebratingresult.

Theorem 4.6. LetG be a fullerene graph with n vertices. Then, ϕ(G) ≤√

125 n. Moreover,

the equality holds if and only if G is an (i, i)-icosahedral fullerene graph.

The bipartite edge frustration is considered for a possible predictor of molecular stabil-ity: If ϕ(G) is close to the upper bound, it means that there is no way to partition the 12pentagonal faces into six pairs in such a way that the pentagons within the pairs are closeto each other.

On the other side, a small value for ϕ(G) does not say much about the structure of thefullerene graph and about the shape of the molecule. The smallest possible value of ϕ(G)is attained for any fullerene graph where the 12 pentagonal faces can be partitioned into 6pairs, adjacent to each other. Those pairs of pentagonal faces can be distributed in a widelyvaried list of possible configurations, leading to molecules of different shapes, includingnanotubes of all possible types.

4.4 Independence number

Another invariant tested as a possible stability predictor is the independence number [42].A set I ⊆ V (G) is independent if no two vertices from I are adjacent inG. The cardinalityof any largest independent set in G is called the independence number of G and denotedby α(G).


Independence number of fullerene graphs attracted attention also in the context of studyof independent sets as possible models for addition of bulky segregated groups such asfree radicals or halogen atoms [10]. Sharp upper bounds on the independence number ofn/2 − 2 for general fullerenes and n/2 − 4 for those with isolated pentagons follow bysimple counting argument [44]. Lower bounds were gradually improved from (almost)trivial α(G) ≥ n/3 valid for all 3-chromatic graphs to α(G) ≥ 3n/8 [63] valid for alltriangle-free planar cubic graphs. A better lower bound of type α(G) ≥ n/2 − C

√n,

where C is some constant, was first established for icosahedral fullerenes [58].This observation was formalized in a pair of conjectures in a recent Ph.D. thesis by

Daugherty ([25, pp. 96]). The first one states that the minimum possible independencenumber is achieved on the icosahedral fullerenes that also figure prominently in Theo-rem 4.6. The second one [25, Conjecture 5.5.2] states the precise form of the conjecturedlower bound, i.e., states that α(G) ≥ n/2 − 3

√n/15. Notice that the constant 3/

√15 is

exactly one half of the constant√

12/5 from Theorem 4.6.The first result that approaches the conjecture is due to Andova et al. [2]. They showed

that α(G) ≥ n/2− 78.58√n for a graph G on n vertices. Finally, using Theorem 4.6, this

conjecture was confirmed [43].

Theorem 4.7. Let G be a fullerene graph with n vertices. Then,

α(G) ≥ n

2−√

3

5n .

Moreover, the equality holds if and only if G is an (i, i)-icosahedral fullerene graph.

Similarly to the bipartite edge frustration, a small value of independence number meansthat the corresponding polyhedron/molecule is close to being of a spherical shape. On theother hand, a value of α(G) close to n/2 does not say much about the structure or the shapeof the molecule.

The relations between diameter and the independence number of fullerenes appearin Conjecture 912 of Graffiti [49]. This conjecture was first established in [2] for largefullerenes using Theorem 4.5 and the lower bound on the independence number estab-lished in [2]. Later this theorem was solved by Faria et al. [51] for all fullerenes, whichconfirms the following nice relation.

Theorem 4.8. If G is a fullerene graph, then

α(G) ≥ 2(diam(G)− 1).

4.5 Spectra of fullerene graphs

An eigenvalue of a graph G is an eigenvalue of its adjacency matrix A(G). The set of alleigenvalues of a graph is called its spectrum. We denote the eigenvalues of G by λ1 ≥λ2 ≥ · · · ≥ λn. The fullerene graph G is 3-regular, and therefore the largest eigenvalue λ1is always equal to 3. Since the fullerene graphs are not bipartite, the smallest eigenvalue isgreater than−3. In this section, we consider the bounds for the smallest eigenvalue λn andthe second largest eigenvalue of a fullerene graph.

The fullerenes can be arranged in decreasing order by their smallest eigenvalue. Thisarrangement is known as the smallest-eigenvalue order. It should be noted that there arenonisomorphic fullerenes with the same smallest eigenvalue. Fowler et al. [51] gave thefirst several fullerene graphs of the smallest-eigenvalue order.

370 Ars Math. Contemp. 11 (2016) 353–379

Theorem 4.9. Amongst all fullerenes the smallest eigenvalue is maximal for the dodeca-hedron.

The eigenvalues of the dodecahedron [24] are 3,√

5, 1, 0,−2,−√

5 with the multiplic-ity, 1, 3, 5, 4, 4, 3, respectively, thus the smallest eigenvalue is−

√5. Observe that there are

unique fullerene graphs on 24 and 26 vertices, while there are two non-isomorphic fullerenegraphs on 28 vertices, one of which is called tetrahedral fullerene. The smallest eigenvalueof these three fullerene graphs is −1−

√2 with multiplicity 2, 3, and 4 respectively.

Theorem 4.10. The smallest eigenvalue of a fullerene with at least 28 vertices, which isnot isomorphic to the tetrahedral fullerene on 28 vertices is less than −1−

√2.

The smallest eigenvalue of the second non-tetrahedral fullerene on 28 vertices is ≈−2.5247.

The above results state that the dodecahedron is the first fullerene in the smallest-eigenvalue order of fullerenes, followed by the unique fullerenes on 24 and 26 verticesand the tetrahedral fullerene on 28 vertices. In the class of isolated pentagon fullerenes thebuckminsterfullerene C60 has the smallest eigenvalue [51].

Theorem 4.11. The icosahedral C60 has maximum smallest eigenvalue amongst all iso-lated pentagon fullerenes.

This result lead to the following result [36]

Theorem 4.12. Among all fullerene graphs with at least 60 vertices, the buckminster-fullerene has the maximum smallest eigenvalue.

Theorem 4.6 and Theorem 4.12 are connected by a result on Laplacian eigenvaluesfrom the monograph by Godsil and Royle ([52, pp. 293]).

Theorem 4.13. Let G be a graph with n vertices. Then bip(G) ≤ n4µ∞(G).

Here bip(G) denotes the maximum number of edges in a bipartite spanning subgraphof G (hence the number of edges in G minus the bipartite edge frustration), and µ∞(G) isthe largest Laplacian eigenvalue of G.

Recall that the smallest eigenvalue λn(G) of a 3-regular graph G and the largest Lapla-cian eigenvalue µ∞(G) of G are related by the following relation ([52, pp. 280]):

λn(G) = 3− µ∞(G).

By plugging this into Theorem 4.13 and noting that bip(G) = 3n/2n−ϕ(G) we obtain anupper bound on λn(G) in terms of the bipartite edge frustration of G of the form λn(G) ≤−3 + 4ϕ(G)/n. By taking into account the upper bound of ϕ(G) from Theorem 4.6 weimmediately obtain the following upper bound on the smallest eigenvalue of a fullerenegraph with n vertices [43].

Theorem 4.14. Let G be a fullerene graph with n vertices. Then,

λn(G) ≤ −3 + 8

√3

5n .


The difference s(G) = 3 − λ2 is known as the separator of G or as a spectral cap.Caporossi and Stevanovic [89] showed that the separator 3 − λ2 of a fullerene graph isat most 1. Even more, using the well known interlacing theorem about the spectra of amatrix, they proved that the separator of a fullerene with n vertices is at most 1 − 3/n.These two results were conjectured by Graffiti. Related to this Doslic and Reti [34] provedthe following theorem:

Theorem 4.15. Let G be a fullerene graph on n vertices. Then, the separator of G is atmost 24/n.

5 Matchings5.1 Number of perfect matchings

Since all carbon atoms are 4-valent, for every atom precisely one of the three bonds shouldbe doubled. Such a set of double bonds is called a Kekule structure in a fullerene. Itcorresponds to the notion of perfect matchings in fullerene graphs: a matching in a graphG is a set of edges of G such that no two edges in M share an end-vertex. A matching Mis perfect if any vertex of G is incident with an edge of M .

The computation of the number of perfect matchings in typical fullerene graphs with asmall number of vertices indicates that this number should grow exponentially with n [30].The first general lower bounds for the number of perfect matchings in fullerene graphs werelinear in the number of vertices [26, 27, 99]. The best of them asserts that a fullerene graphwith n vertices has at least d3(n+ 2)/4e different perfect matchings [99]. The numberof perfect matchings was proved to be exponential for several special classes of fullerenegraphs: (0, 5)-nanotubes [77], icosahedral fullerenes [30] or those who are obtained usingspecific operations [32], before the following general result from [69].

Theorem 5.1. Let G be a fullerene graph with n vertices that has no non-trivial cyclic5-edge-cut. The number of perfect matchings of G is at least 2

n−38061 .

The idea of the proof is to find a perfect matching with (n − 380)/61 non-adjacentresonant hexagons. For such a perfect matching, any subset of the set of resonant hexagonscan be switched to obtain a different perfect matching. The bound from Theorem 5.1 is notoptimal. Doslic [33] studied the number of perfect matchings in leapfrog fullerene graphs,and found the following.

Theorem 5.2. Let G be a leapfrog fullerene on n vertices. If n is divisible by 4, then thenumber of perfect matchings is at least Fn/4+1 + n/2, otherwise the number of perfectmatchings is at least Fdn/4e+1 + 1, where Fn denotes the n-th Fibonacci number.

A little is known about the relationship between the number of perfect matchings of afullerene graph and the shape of the corresponding molecule.

5.2 Resonance in fullerene graphs

Let M be a perfect matching in a fullerene graph G. A hexagonal face is resonant if it isincident with three edges in M . A set of disjoint resonant hexagons of G is called resonantpattern (also known as sextet pattern). A fullerene graph is k-resonant if any i, 0 < i ≤ k,disjoint hexagons form a resonant pattern.

372 Ars Math. Contemp. 11 (2016) 353–379

Ye et al. [94] showed that every fullerene graph is 1-resonant. Even more, they provedthat every fullerene leapfrog graph is 2-resonant. Since every leapfrog fullerene graph isIPR fullerene, they naturally ask if all IPR fullerenes are 2-resonant. Kaiser et al. [67]answered this question.

Theorem 5.3. Every IPR fullerene graph is 2-resonant.

The fullerene graphs in general are not 3-resonant, i.e., only “small” fullerenes are3-resonant [94].

Theorem 5.4. If a fullerene graph G is 3-resonant, then |V (G)| ≤ 60.

The maximum size of a set of resonant hexagons in G is called the Clar number of G,and denoted by c(G). Zhang and Ye [98] showed the following.

Theorem 5.5. A Clar number on fullerene graph with n vertices is at most⌊n− 12

6

⌋.

They also showed that there are infinitely many fullerene graphs that attain this upperbound; among all 1812 fullerenes on 60 vertices there are exactly 18 whose Clar numberis 8 (including the buckminsterfullerene). This upper bound is achieved for C70, and forinfinitely many armchair and zig-zag nanotubes.

The sextet polynomial is defined in [64, 85] as BG(x) =∑m

k=0 r(G, k)xk, wherer(G, k) is the number of resonant patterns of G of size k, and m is the Clar number ofG. Sereni and Stehlık [88] proved a conjecture made by Zhang and He [97] that for everyfullerene graph the value BG(1) is smaller than the number of perfect matchings in thegraph.

A k-regular spanning subgraph of a graph G is called k-factor of G. A fullerene thathas a 2-factor form by the boundaries of it faces is called fullerene of Clar type. It is knownthat if a fullerene is of Clar type, then it has isolated pentagons, i.e., it is a IPR fullerene,see Pisanski [86]. It also holds that the number of fullerenes with n vertices is equal to thenumber of Clar type fullerenes with 3n vertices.

5.3 Saturation number

The last invariant considered here, the saturation number, is also related to matchings.The existence of perfect (and hence maximum) matchings in fullerene graphs has beenestablished long time ago, and there are many papers concerned with their structural andenumerative properties [27, 69, 72]. Another class of matchings, the maximal matchings,have received much less attention so far, in spite of being potentially useful as mathematicalmodels of dimer absorption. A matching M is maximal if it cannot be extended to a largermatching of G. The saturation number of G, denoted by s(G), is the cardinality of anysmallest maximal matching ofG. The saturation number of fullerene graphs was studied in[27, 36]. Using the lower bound on the diameter the bounds on the saturation number wereimproved in [2]. Still that was not the best possible bound of the saturation number of thefullerene graphs. In [3] we prove that the saturation number for fullerenes on n vertices isessentially n/3.

Theorem 5.6. Let F be a fullerene graph on n vertices. Then,

n

3− 2 ≤ s(F ) ≤ n

3+O(

√n) .


In order to prove the lower bound of this theorem we used the discharging method. Forthe upper bound we first used Theorem 4.6, and obtained a bipartite graph F ′. Later, weestablished that F ′ is an induced subgraph of a hexagonal lattice, or an induced subgraphof a hexagonal tube (defined as on Figure 6). Then we defined a maximal matching on F ′

such that from each hexagon precisely four vertices are covered by the matching.

6 Hamiltonicity of fullerene graphsA cycle C of a graph G is hamiltonian if C passes through every vertex of G. Tait [91] in1884 conjectured that every 3-connected planar cubic graph has a Hamiltonian cycle. LaterTutte [92] showed that this conjecture is false. Back in 1974, Grunbaum and Zaks [60]asked whether 3-connected planar graphs with faces of size p and q, q > p are Hamiltonianfor any p and q. It was shown that cubic polyhedral graphs with faces of sizes 3 and 6, orfaces of size 4 and 6 are hamiltonian [55, 56].

However, Tait’s Conjecture still remained open for fullerene graphs. Barnette [9] madea more general conjecture, he conjectured that all cubic polyhedral graphs with maximumface size at most six having no triangles and no two adjacent quadrangles are hamiltonian.He reduced the conjecture to the case of cubic polyhedral graphs with maximum face sizeat most six with no triangles nor two adjacent quadrangles, are also known as Barnettegraphs. In order to attack the hamiltonicity conjecture for fullerene graphs, long cyclesin fullerenes were studied [40, 66, 74, 39]. Marusic [82] showed that for every fullerenegraph, its leapfrog fullerene contains a hamiltonian path. Kardos [68] on the other sideworked on the more general conjecture, Barnette’s Conjecture, and proved the following.

Theorem 6.1. Let G be a 3-connected planar cubic graph with faces of size at most 6.Then G is hamiltonian.

This theorem directly implies implies Taits’s Conjecture on fullerene graphs. For theproof of this theorem a more general definition of a 2-factor was introduced, in this case a2-factor is any spanning subgraph F of a Barnette graph G such that each component ofF is a connected regular graph of degree 1 or 2, i.e., an isolated edge (considered here as a2-cycle) or a cycle. The proof has several major steps. First a 2-factor with many resonanthexagons is found. Then the 2-factor is modified in order to have an odd number of cyclesif needed. Finally the resonant hexagons are used to glue all the cycles into a single cycle,i.e., to construct a Hamiltonian cycle. In order to achieve this, a computer program wasused.

7 Fullerene graphs and topological indicesTopological indices or molecular descriptors are graph invariants than can correlate witha physical or chemical property of a molecule presented as a graph. The first topologicalindex was introduced by Harry Wiener [93], back in 1947, and afterwards a list of descrip-tors were introduced such as: the Randic index, the Hosoya index, the Estrada index, theZagreb Indices, the Szeged index, the Balaban index, etc.

Let G be a graph and u ∈ V (G). Then the distance of u is defined as WG(u) =∑v∈V (G) dG(u, v) , where d(u, v) denotes the distance between the vertices u and v.

The Wiener index [93] of a graph G, W (G), is the sum of all the distances in G, i.e.,W (G) =

∑u,v∈V (G) d(u, v) = 1

2

∑u∈V (G)WG(u). Hua et al. [65] considered a Wiener

374 Ars Math. Contemp. 11 (2016) 353–379

dimension of one type of (0, 6)-nanotubes, and based on the results they made the followingconjecture.

Conjecture 7.1. Let G be a fullerene with n vertices. Then, the Wiener index of G, W (G),is of order Θ(n3).

The bound of the Wiener index for nanotubical fullerenes are determined in [4]. Sup-pose that WG(u) | u ∈ V (G) = d1, d2, . . . , dk. Assume in addition that G containsti pairs of vertices on distance di, 1 ≤ i ≤ k. We say that the Wiener dimension dimW(G)of G is k. Alizadeh et al. [1] have shown that the (0, 5)-nanotubical fullerene graph on 10k(k ≥ 3) vertices has the Wiener dimension k. As a consequence the Wiener index of thesefullerenes was obtained. This result confirms Conjecture 7.1 for (0, 5)-nanotubes.

Another topological index based on the distance is the Balaban index. This index for agraph G is defined [7] as

J(G) =m

m− n+ 2

∑u,v∈E(G)

1√dG(u) · dG(v)

,

where n and m are the number of vertices and edges, respectively.Knor et al. [73] studied the Balaban index for cubic graphs, and determined that the

Balaban index tends to zero as the number of vertices increases. More precisely, theyshowed the following.

Theorem 7.2. Let G be a a fullerene graph on n vertices, n ≥ 60. Then J(G) ≤ 25/√n.

The last result means that Balaban index does not distinguish well the fullerene graphswhen they are sufficiently large. Determining the distances on an infinite hexagonal tube [5]the authors [4] show that this approach is much faster for nanotubes.

Let G be a fullerene graph and e = uv an edge in G. Let nv(e) be the number ofedges closer to v than to the vertex u. Similarly, let n0(e) be the number of equidistantvertices form u and v. The following indices are base on these invariants. The PI indexis defined by PI(G) =

∑e=uv[nv(e) + nu(e)]. The Szeged index is topological index

introduced by Gutman [61], defined as Sz(G) =∑

e=uv nv(e)nu(e). Randic [87] modifiedthis topological index in order to find better applications in chemistry, and the new indexwas named revised Szeged index. This index is defined by Sz∗(G) =

∑e=uv[nv(e) +

n0(e)/2]·[nu(e)+n0(e)/2]. Mottaaghi and Mehranian [83] studied PI, Szeged and revisedSzeged index for a nanotubical IPR fullerenes on 60+12n, n ∈ N, vertices, and found thatthese indices are of order Θ(n2), Θ(n3) and Θ(n3), respectively.

8 Fullerene generating programsThere are several programs available that can generate fullerenes. Buckygen is a programby Brinkmann, McKay and Goedgebeur for the efficient generation of all nonisomorphicfullerenes. It generates triangulations where all vertices have degree 5 or 6, or its dualrepresentation-fullerene graphs. This program can also be used to generate isolated pen-tagon rule (IPR) fullerenes. SaGe generates fullerene graphs using the Buckygen generator.The algorithms used in the Buckygen generator are described in [16, 53]. Programs forgeneration of certain types of graphs are Plantri and Fullgen, both created by Brinkmannand McKay. These programs are based on the papers [17, 19, 20]. GaGe [18] is an Open


Source program implemented in C and Java, that generates graphs of different types. Thisprogram allows to view selected graphs in various ways or save them in several formats.House of Graphs provides a count lists of fullerene, fullerenes without a spiral starting atpentagon, and fullerenes without a spiral. These fullerenes are generated with Buckygenand Fullgen.

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Mathematical aspects of fullerenes